Introduction to Differentiation

The gradient of a curve $y=f(x)$ at a specific point measures its steepness or instantaneous rate of change at that point. Geometrically, it's the slope of the line tangent to the curve at that point.

The Secant Line

To understand this, consider two points on the curve: $P(x, f(x))$ and a nearby point $Q(x+h, f(x+h))$. The value $h$ represents a small change in $x$. The straight line connecting $P$ and $Q$ is called a secant line. Its slope, $m_{PQ}$, is given by the familiar rise-over-run formula:

$$ m_{PQ} = \frac{\text{change in y}}{\text{change in x}} = \frac{f(x+h) - f(x)}{(x+h) - x} = \frac{f(x+h) - f(x)}{h} $$

This expression is also known as the difference quotient.

The Tangent Line and the Limit

To find the gradient precisely at point $P$, we imagine moving point $Q$ closer and closer to $P$. As $Q$ approaches $P$, the distance $h$ approaches zero. The secant line $PQ$ then approaches the tangent line at $P$.

The gradient of the curve at $P$ is defined as the limiting value of the slope of this secant line as $h \to 0$. This limit, if it exists, is the derivative of $f(x)$ with respect to $x$, denoted as $f'(x)$ or $\frac{dy}{dx}$.

This leads to the fundamental definition of the derivative, often called differentiation from first principles: $$ f'(x) = \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$ This definition forms the bedrock of differential calculus, allowing us to find the instantaneous rate of change of a function.